Properties

Label 8001.2.a.t
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{11} ) q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{11} ) q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{13} ) q^{11} + ( \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} -\beta_{1} q^{14} + ( -1 - \beta_{2} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{16} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{17} + ( -2 - \beta_{2} + \beta_{8} - \beta_{11} + 2 \beta_{14} ) q^{19} + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{20} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{22} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} - 2 \beta_{15} ) q^{25} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{14} ) q^{29} + ( -1 - \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{31} + ( -1 - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{34} + ( -1 + \beta_{11} ) q^{35} + ( \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{37} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{38} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{40} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{15} ) q^{41} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{43} + ( 1 + \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{44} + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{46} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{47} + q^{49} + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{9} - 4 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{50} + ( 1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{52} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{53} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{56} + ( 1 + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{58} + ( 5 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 4 - \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{61} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{62} + ( -3 - 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{64} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{65} + ( 2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{67} + ( 4 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{68} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{70} + ( 6 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{13} ) q^{73} + ( 4 + 3 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{74} + ( \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{76} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{77} + ( -6 - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{79} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{80} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{82} + ( 6 + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{83} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{85} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{86} + ( -4 + 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 6 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} + 4 \beta_{15} ) q^{88} + ( 4 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( -\beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{91} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 4 \beta_{14} ) q^{92} + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{94} + ( -\beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} + 3 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} + ( 2 + 3 \beta_{2} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{2} + 12q^{4} + 9q^{5} - 16q^{7} + 6q^{8} + O(q^{10}) \) \( 16q + 2q^{2} + 12q^{4} + 9q^{5} - 16q^{7} + 6q^{8} - 2q^{10} + 22q^{11} - 4q^{13} - 2q^{14} + 12q^{16} + 18q^{17} - 15q^{19} + 40q^{20} - 11q^{22} + 5q^{23} + 15q^{25} + 24q^{26} - 12q^{28} + 12q^{29} - 32q^{31} + 9q^{32} - 14q^{34} - 9q^{35} - 2q^{37} - 3q^{38} - 14q^{40} + 45q^{41} - 3q^{43} + 54q^{44} + 49q^{47} + 16q^{49} + 6q^{50} + 38q^{52} - 16q^{53} + 7q^{55} - 6q^{56} + 16q^{58} + 35q^{59} - 11q^{61} - 17q^{62} - 2q^{64} - 14q^{65} + 17q^{67} + 71q^{68} + 2q^{70} + 81q^{71} - 15q^{73} - 13q^{74} + 14q^{76} - 22q^{77} - 34q^{79} + 33q^{80} - 14q^{82} + 39q^{83} - 17q^{85} - 36q^{86} + 61q^{88} + 32q^{89} + 4q^{91} - 37q^{92} + 13q^{94} + 33q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-116093 \nu^{15} + 228893 \nu^{14} + 2347523 \nu^{13} - 4412026 \nu^{12} - 18385016 \nu^{11} + 32533154 \nu^{10} + 70696203 \nu^{9} - 115867538 \nu^{8} - 138531569 \nu^{7} + 205570931 \nu^{6} + 126910371 \nu^{5} - 162346281 \nu^{4} - 41913198 \nu^{3} + 34856907 \nu^{2} + 6759511 \nu - 128712\)\()/239255\)
\(\beta_{4}\)\(=\)\((\)\(118542 \nu^{15} - 282272 \nu^{14} - 2211727 \nu^{13} + 5170999 \nu^{12} + 15614719 \nu^{11} - 35484771 \nu^{10} - 52971862 \nu^{9} + 114412902 \nu^{8} + 91508371 \nu^{7} - 178289744 \nu^{6} - 79215634 \nu^{5} + 121751459 \nu^{4} + 32149637 \nu^{3} - 26089138 \nu^{2} - 4109094 \nu + 299938\)\()/239255\)
\(\beta_{5}\)\(=\)\((\)\(246643 \nu^{15} - 556793 \nu^{14} - 4734663 \nu^{13} + 10433381 \nu^{12} + 34632721 \nu^{11} - 74008394 \nu^{10} - 121920963 \nu^{9} + 250361358 \nu^{8} + 213453449 \nu^{7} - 416685796 \nu^{6} - 167476866 \nu^{5} + 306694906 \nu^{4} + 40444858 \nu^{3} - 64193652 \nu^{2} - 2974546 \nu + 1167877\)\()/239255\)
\(\beta_{6}\)\(=\)\((\)\(-416133 \nu^{15} + 848713 \nu^{14} + 8205093 \nu^{13} - 16039026 \nu^{12} - 62183706 \nu^{11} + 115239599 \nu^{10} + 229254488 \nu^{9} - 397334578 \nu^{8} - 425687974 \nu^{7} + 679630526 \nu^{6} + 361459511 \nu^{5} - 519191331 \nu^{4} - 101394473 \nu^{3} + 112731547 \nu^{2} + 10770031 \nu - 1601912\)\()/239255\)
\(\beta_{7}\)\(=\)\((\)\(-420868 \nu^{15} + 831558 \nu^{14} + 8486798 \nu^{13} - 16040641 \nu^{12} - 66156781 \nu^{11} + 118505764 \nu^{10} + 252322388 \nu^{9} - 423867643 \nu^{8} - 486918174 \nu^{7} + 758894461 \nu^{6} + 431479801 \nu^{5} - 611916381 \nu^{4} - 127789533 \nu^{3} + 142341487 \nu^{2} + 15316331 \nu - 2488852\)\()/239255\)
\(\beta_{8}\)\(=\)\((\)\(86516 \nu^{15} - 193256 \nu^{14} - 1686213 \nu^{13} + 3671187 \nu^{12} + 12597346 \nu^{11} - 26550396 \nu^{10} - 45669957 \nu^{9} + 92287029 \nu^{8} + 83323677 \nu^{7} - 159344793 \nu^{6} - 69683347 \nu^{5} + 123008780 \nu^{4} + 19521756 \nu^{3} - 27300111 \nu^{2} - 2391162 \nu + 403609\)\()/47851\)
\(\beta_{9}\)\(=\)\((\)\(-558146 \nu^{15} + 1242406 \nu^{14} + 10825416 \nu^{13} - 23508057 \nu^{12} - 80263942 \nu^{11} + 169127043 \nu^{10} + 287448331 \nu^{9} - 583976571 \nu^{8} - 513565088 \nu^{7} + 1000551217 \nu^{6} + 412351657 \nu^{5} - 766742622 \nu^{4} - 104612936 \nu^{3} + 169828364 \nu^{2} + 14520502 \nu - 2398759\)\()/239255\)
\(\beta_{10}\)\(=\)\((\)\(609731 \nu^{15} - 1339991 \nu^{14} - 11975331 \nu^{13} + 25605932 \nu^{12} + 90323612 \nu^{11} - 186671853 \nu^{10} - 331094336 \nu^{9} + 655602681 \nu^{8} + 610664783 \nu^{7} - 1145738667 \nu^{6} - 514021097 \nu^{5} + 894448927 \nu^{4} + 142265856 \nu^{3} - 197861309 \nu^{2} - 18219637 \nu + 2939939\)\()/239255\)
\(\beta_{11}\)\(=\)\((\)\(-614167 \nu^{15} + 1330387 \nu^{14} + 12063962 \nu^{13} - 25378599 \nu^{12} - 90934424 \nu^{11} + 184614846 \nu^{10} + 332488247 \nu^{9} - 646723747 \nu^{8} - 608767076 \nu^{7} + 1127337289 \nu^{6} + 501713704 \nu^{5} - 878849284 \nu^{4} - 127489287 \nu^{3} + 195153183 \nu^{2} + 14029589 \nu - 2942448\)\()/239255\)
\(\beta_{12}\)\(=\)\((\)\(-673264 \nu^{15} + 1527129 \nu^{14} + 13105079 \nu^{13} - 29129163 \nu^{12} - 97629743 \nu^{11} + 211857962 \nu^{10} + 351724054 \nu^{9} - 742060049 \nu^{8} - 632322762 \nu^{7} + 1294215363 \nu^{6} + 508299188 \nu^{5} - 1013016703 \nu^{4} - 121901624 \nu^{3} + 232294846 \nu^{2} + 12764513 \nu - 5229541\)\()/239255\)
\(\beta_{13}\)\(=\)\((\)\(907437 \nu^{15} - 1921932 \nu^{14} - 18024552 \nu^{13} + 36966909 \nu^{12} + 137780884 \nu^{11} - 271915646 \nu^{10} - 512234712 \nu^{9} + 966370827 \nu^{8} + 954511391 \nu^{7} - 1714237034 \nu^{6} - 797268019 \nu^{5} + 1362704444 \nu^{4} + 198561717 \nu^{3} - 308068303 \nu^{2} - 20676569 \nu + 5318248\)\()/239255\)
\(\beta_{14}\)\(=\)\((\)\(-183898 \nu^{15} + 398886 \nu^{14} + 3621479 \nu^{13} - 7620756 \nu^{12} - 27413174 \nu^{11} + 55548928 \nu^{10} + 100969347 \nu^{9} - 195111036 \nu^{8} - 187405313 \nu^{7} + 341269962 \nu^{6} + 159092271 \nu^{5} - 267285935 \nu^{4} - 44412449 \nu^{3} + 59983859 \nu^{2} + 5138082 \nu - 1000302\)\()/47851\)
\(\beta_{15}\)\(=\)\((\)\(929511 \nu^{15} - 2103041 \nu^{14} - 18058381 \nu^{13} + 39940442 \nu^{12} + 134317747 \nu^{11} - 288743733 \nu^{10} - 483981381 \nu^{9} + 1003060396 \nu^{8} + 875314583 \nu^{7} - 1730295122 \nu^{6} - 721664127 \nu^{5} + 1334031662 \nu^{4} + 195849826 \nu^{3} - 296868004 \nu^{2} - 24599592 \nu + 5430944\)\()/239255\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{15} + 2 \beta_{14} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 5 \beta_{2} + 13\)
\(\nu^{5}\)\(=\)\(9 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + 10 \beta_{12} + \beta_{11} + \beta_{10} - 9 \beta_{5} + 10 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 28 \beta_{1} - 9\)
\(\nu^{6}\)\(=\)\(10 \beta_{15} + 22 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - 7 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + 13 \beta_{8} - 10 \beta_{7} - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 24 \beta_{2} + 63\)
\(\nu^{7}\)\(=\)\(68 \beta_{15} + 72 \beta_{14} + 81 \beta_{13} + 81 \beta_{12} + 15 \beta_{11} + 16 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + \beta_{6} - 66 \beta_{5} + 80 \beta_{4} + 107 \beta_{3} - 70 \beta_{2} + 166 \beta_{1} - 71\)
\(\nu^{8}\)\(=\)\(81 \beta_{15} + 193 \beta_{14} + 49 \beta_{13} + 45 \beta_{12} - 30 \beta_{11} + 46 \beta_{10} + 64 \beta_{9} + 124 \beta_{8} - 81 \beta_{7} - 16 \beta_{5} + 35 \beta_{4} + 92 \beta_{3} + 109 \beta_{2} + \beta_{1} + 320\)
\(\nu^{9}\)\(=\)\(492 \beta_{15} + 555 \beta_{14} + 618 \beta_{13} + 614 \beta_{12} + 161 \beta_{11} + 171 \beta_{10} - 14 \beta_{9} + 49 \beta_{8} - 45 \beta_{7} + 12 \beta_{6} - 460 \beta_{5} + 599 \beta_{4} + 864 \beta_{3} - 517 \beta_{2} + 1021 \beta_{1} - 537\)
\(\nu^{10}\)\(=\)\(619 \beta_{15} + 1562 \beta_{14} + 537 \beta_{13} + 469 \beta_{12} - 45 \beta_{11} + 490 \beta_{10} + 428 \beta_{9} + 1041 \beta_{8} - 614 \beta_{7} - 2 \beta_{6} - 172 \beta_{5} + 398 \beta_{4} + 969 \beta_{3} + 448 \beta_{2} + 22 \beta_{1} + 1658\)
\(\nu^{11}\)\(=\)\(3511 \beta_{15} + 4202 \beta_{14} + 4602 \beta_{13} + 4524 \beta_{12} + 1495 \beta_{11} + 1567 \beta_{10} - 134 \beta_{9} + 549 \beta_{8} - 469 \beta_{7} + 96 \beta_{6} - 3169 \beta_{5} + 4376 \beta_{4} + 6668 \beta_{3} - 3728 \beta_{2} + 6447 \beta_{1} - 3961\)
\(\nu^{12}\)\(=\)\(4639 \beta_{15} + 12147 \beta_{14} + 5003 \beta_{13} + 4231 \beta_{12} + 822 \beta_{11} + 4497 \beta_{10} + 2816 \beta_{9} + 8185 \beta_{8} - 4524 \beta_{7} - 46 \beta_{6} - 1592 \beta_{5} + 3796 \beta_{4} + 8793 \beta_{3} + 1437 \beta_{2} + 322 \beta_{1} + 8626\)
\(\nu^{13}\)\(=\)\(24939 \beta_{15} + 31494 \beta_{14} + 33856 \beta_{13} + 32868 \beta_{12} + 12804 \beta_{11} + 13298 \beta_{10} - 1092 \beta_{9} + 5271 \beta_{8} - 4231 \beta_{7} + 631 \beta_{6} - 21848 \beta_{5} + 31652 \beta_{4} + 50232 \beta_{3} - 26569 \beta_{2} + 41544 \beta_{1} - 28730\)
\(\nu^{14}\)\(=\)\(34545 \beta_{15} + 92328 \beta_{14} + 42883 \beta_{13} + 35508 \beta_{12} + 13085 \beta_{11} + 38161 \beta_{10} + 18504 \beta_{9} + 62007 \beta_{8} - 32868 \beta_{7} - 653 \beta_{6} - 13716 \beta_{5} + 33079 \beta_{4} + 74013 \beta_{3} + 702 \beta_{2} + 3857 \beta_{1} + 44462\)
\(\nu^{15}\)\(=\)\(176958 \beta_{15} + 234588 \beta_{14} + 247371 \beta_{13} + 237044 \beta_{12} + 104286 \beta_{11} + 107926 \beta_{10} - 8127 \beta_{9} + 46678 \beta_{8} - 35508 \beta_{7} + 3571 \beta_{6} - 151318 \beta_{5} + 228008 \beta_{4} + 373091 \beta_{3} - 188317 \beta_{2} + 272149 \beta_{1} - 206072\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.57541
−2.24781
−1.72082
−1.48078
−1.45188
−0.532475
−0.191617
−0.170601
0.102239
0.625678
1.24549
1.33134
1.79993
2.18919
2.37299
2.70451
−2.57541 0 4.63271 3.61540 0 −1.00000 −6.78030 0 −9.31112
1.2 −2.24781 0 3.05263 0.411504 0 −1.00000 −2.36611 0 −0.924982
1.3 −1.72082 0 0.961212 4.11468 0 −1.00000 1.78756 0 −7.08062
1.4 −1.48078 0 0.192706 −1.52223 0 −1.00000 2.67620 0 2.25409
1.5 −1.45188 0 0.107956 −0.584615 0 −1.00000 2.74702 0 0.848791
1.6 −0.532475 0 −1.71647 0.118172 0 −1.00000 1.97893 0 −0.0629236
1.7 −0.191617 0 −1.96328 −3.92161 0 −1.00000 0.759431 0 0.751445
1.8 −0.170601 0 −1.97090 0.206353 0 −1.00000 0.677439 0 −0.0352041
1.9 0.102239 0 −1.98955 −0.280585 0 −1.00000 −0.407889 0 −0.0286868
1.10 0.625678 0 −1.60853 0.920707 0 −1.00000 −2.25778 0 0.576066
1.11 1.24549 0 −0.448744 2.43368 0 −1.00000 −3.04990 0 3.03114
1.12 1.33134 0 −0.227522 2.92411 0 −1.00000 −2.96560 0 3.89299
1.13 1.79993 0 1.23975 −3.74624 0 −1.00000 −1.36840 0 −6.74297
1.14 2.18919 0 2.79257 −0.868150 0 −1.00000 1.73509 0 −1.90055
1.15 2.37299 0 3.63108 3.84175 0 −1.00000 3.87053 0 9.11644
1.16 2.70451 0 5.31438 1.33706 0 −1.00000 8.96376 0 3.61609
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.t 16
3.b odd 2 1 889.2.a.c 16
21.c even 2 1 6223.2.a.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.c 16 3.b odd 2 1
6223.2.a.k 16 21.c even 2 1
8001.2.a.t 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{16} - \cdots\)
\(T_{5}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 12 T^{2} - 22 T^{3} + 75 T^{4} - 127 T^{5} + 327 T^{6} - 517 T^{7} + 1117 T^{8} - 1665 T^{9} + 3198 T^{10} - 4555 T^{11} + 8050 T^{12} - 11054 T^{13} + 18325 T^{14} - 24255 T^{15} + 38213 T^{16} - 48510 T^{17} + 73300 T^{18} - 88432 T^{19} + 128800 T^{20} - 145760 T^{21} + 204672 T^{22} - 213120 T^{23} + 285952 T^{24} - 264704 T^{25} + 334848 T^{26} - 260096 T^{27} + 307200 T^{28} - 180224 T^{29} + 196608 T^{30} - 65536 T^{31} + 65536 T^{32} \)
$3$ 1
$5$ \( 1 - 9 T + 73 T^{2} - 402 T^{3} + 1978 T^{4} - 8015 T^{5} + 29678 T^{6} - 96727 T^{7} + 295846 T^{8} - 831141 T^{9} + 2261906 T^{10} - 5860568 T^{11} + 15073267 T^{12} - 37453319 T^{13} + 92144207 T^{14} - 216794219 T^{15} + 497747736 T^{16} - 1083971095 T^{17} + 2303605175 T^{18} - 4681664875 T^{19} + 9420791875 T^{20} - 18314275000 T^{21} + 35342281250 T^{22} - 64932890625 T^{23} + 115564843750 T^{24} - 188919921875 T^{25} + 289824218750 T^{26} - 391357421875 T^{27} + 482910156250 T^{28} - 490722656250 T^{29} + 445556640625 T^{30} - 274658203125 T^{31} + 152587890625 T^{32} \)
$7$ \( ( 1 + T )^{16} \)
$11$ \( 1 - 22 T + 303 T^{2} - 3067 T^{3} + 25368 T^{4} - 177770 T^{5} + 1090130 T^{6} - 5934857 T^{7} + 29109808 T^{8} - 129561057 T^{9} + 528043225 T^{10} - 1982539376 T^{11} + 6932748788 T^{12} - 22873704378 T^{13} + 72928938613 T^{14} - 231216828675 T^{15} + 752600334649 T^{16} - 2543385115425 T^{17} + 8824401572173 T^{18} - 30444900527118 T^{19} + 101502375005108 T^{20} - 319289949044176 T^{21} + 935460783724225 T^{22} - 2524778472699747 T^{23} + 6239945869004848 T^{24} - 13994082359565187 T^{25} + 28275164680288130 T^{26} - 50719855684517470 T^{27} + 79615651060658328 T^{28} - 105881158145436377 T^{29} + 115064199575722023 T^{30} - 91899459727144322 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 + 4 T + 107 T^{2} + 380 T^{3} + 5789 T^{4} + 19050 T^{5} + 212144 T^{6} + 652425 T^{7} + 5882030 T^{8} + 16921131 T^{9} + 131205020 T^{10} + 352876433 T^{11} + 2446352235 T^{12} + 6149085091 T^{13} + 39093257012 T^{14} + 91927991319 T^{15} + 543536230395 T^{16} + 1195063887147 T^{17} + 6606760435028 T^{18} + 13509539944927 T^{19} + 69870266183835 T^{20} + 131020549437869 T^{21} + 633301571381180 T^{22} + 1061775876212727 T^{23} + 4798152572843630 T^{24} + 6918640503429525 T^{25} + 29245851894814256 T^{26} + 34140655506404850 T^{27} + 134872614774042509 T^{28} + 115092540505056140 T^{29} + 421299273269823923 T^{30} + 204743572056363028 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - 18 T + 273 T^{2} - 2781 T^{3} + 25658 T^{4} - 193733 T^{5} + 1383023 T^{6} - 8697320 T^{7} + 53036891 T^{8} - 294777276 T^{9} + 1607117632 T^{10} - 8112035114 T^{11} + 40363531935 T^{12} - 187473496595 T^{13} + 860633799060 T^{14} - 3703086251987 T^{15} + 15762439781607 T^{16} - 62952466283779 T^{17} + 248723167928340 T^{18} - 921057288771235 T^{19} + 3371202550743135 T^{20} - 11517929840858698 T^{21} + 38791912733516608 T^{22} - 120958516264394748 T^{23} + 369972487040755931 T^{24} - 1031396710014888040 T^{25} + 2788165932180677327 T^{26} - 6639597287366663989 T^{27} + 14948921362841207738 T^{28} - 27544631509511410797 T^{29} + 45967146650716453617 T^{30} - 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 + 15 T + 248 T^{2} + 2524 T^{3} + 25137 T^{4} + 195368 T^{5} + 1456692 T^{6} + 9129794 T^{7} + 54772327 T^{8} + 283776189 T^{9} + 1411840969 T^{10} + 6094322619 T^{11} + 25604146193 T^{12} + 93020585977 T^{13} + 350079022597 T^{14} + 1197314079224 T^{15} + 5166511617041 T^{16} + 22748967505256 T^{17} + 126378527157517 T^{18} + 638028199216243 T^{19} + 3336757936017953 T^{20} + 15090146142583281 T^{21} + 66421302218498689 T^{22} + 253659515548222671 T^{23} + 930229268506766407 T^{24} + 2946072207056527526 T^{25} + 8931075169208654292 T^{26} + 22758468900427249592 T^{27} + 55636097120566089057 T^{28} + \)\(10\!\cdots\!16\)\( T^{29} + \)\(19\!\cdots\!08\)\( T^{30} + \)\(22\!\cdots\!85\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - 5 T + 180 T^{2} - 911 T^{3} + 16850 T^{4} - 85251 T^{5} + 1077221 T^{6} - 5402068 T^{7} + 52468559 T^{8} - 258760441 T^{9} + 2064324615 T^{10} - 9931668180 T^{11} + 68039962889 T^{12} - 316125223782 T^{13} + 1923834329544 T^{14} - 8513441106040 T^{15} + 47326552683658 T^{16} - 195809145438920 T^{17} + 1017708360328776 T^{18} - 3846295597755594 T^{19} + 19040371254820649 T^{20} - 63923622968665740 T^{21} + 305594129566107735 T^{22} - 881034134193742127 T^{23} + 4108864551564280079 T^{24} - 9729949155604105484 T^{25} + 44625507836078189429 T^{26} - 81227984671920190677 T^{27} + \)\(36\!\cdots\!50\)\( T^{28} - \)\(45\!\cdots\!13\)\( T^{29} + \)\(20\!\cdots\!20\)\( T^{30} - \)\(13\!\cdots\!35\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 12 T + 296 T^{2} - 2842 T^{3} + 41236 T^{4} - 340889 T^{5} + 3735779 T^{6} - 27481870 T^{7} + 249794460 T^{8} - 1665448573 T^{9} + 13186445822 T^{10} - 80594686502 T^{11} + 572307864911 T^{12} - 3229196488825 T^{13} + 20940762186163 T^{14} - 109497405605747 T^{15} + 655215411792280 T^{16} - 3175424762566663 T^{17} + 17611180998563083 T^{18} - 78756873165952925 T^{19} + 404782479002116991 T^{20} - 1653089623450810798 T^{21} + 7843605496028614862 T^{22} - 28728781883250557057 T^{23} + \)\(12\!\cdots\!60\)\( T^{24} - \)\(39\!\cdots\!30\)\( T^{25} + \)\(15\!\cdots\!79\)\( T^{26} - \)\(41\!\cdots\!81\)\( T^{27} + \)\(14\!\cdots\!76\)\( T^{28} - \)\(29\!\cdots\!38\)\( T^{29} + \)\(88\!\cdots\!76\)\( T^{30} - \)\(10\!\cdots\!88\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 32 T + 744 T^{2} + 13016 T^{3} + 192404 T^{4} + 2460971 T^{5} + 28126531 T^{6} + 290885953 T^{7} + 2761525270 T^{8} + 24235747979 T^{9} + 198158955757 T^{10} + 1515764606612 T^{11} + 10895848982445 T^{12} + 73780338113067 T^{13} + 471827014000243 T^{14} + 2852663099422432 T^{15} + 16325698746086223 T^{16} + 88432556082095392 T^{17} + 453425760454233523 T^{18} + 2197990052726378997 T^{19} + 10062545348116588845 T^{20} + 43395053803150546412 T^{21} + \)\(17\!\cdots\!17\)\( T^{22} + \)\(66\!\cdots\!69\)\( T^{23} + \)\(23\!\cdots\!70\)\( T^{24} + \)\(76\!\cdots\!63\)\( T^{25} + \)\(23\!\cdots\!31\)\( T^{26} + \)\(62\!\cdots\!01\)\( T^{27} + \)\(15\!\cdots\!44\)\( T^{28} + \)\(31\!\cdots\!56\)\( T^{29} + \)\(56\!\cdots\!24\)\( T^{30} + \)\(75\!\cdots\!32\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 2 T + 264 T^{2} + 460 T^{3} + 36898 T^{4} + 50046 T^{5} + 3563953 T^{6} + 3368000 T^{7} + 264976726 T^{8} + 146604940 T^{9} + 16104532122 T^{10} + 3592350051 T^{11} + 830343504612 T^{12} - 7778681402 T^{13} + 37173486688105 T^{14} - 4566087322258 T^{15} + 1463655345234495 T^{16} - 168945230923546 T^{17} + 50890503276015745 T^{18} - 394013549055506 T^{19} + 1556197412947130532 T^{20} + 249107767465491807 T^{21} + 41319823370004209898 T^{22} + 13917482151170837020 T^{23} + \)\(93\!\cdots\!46\)\( T^{24} + \)\(43\!\cdots\!00\)\( T^{25} + \)\(17\!\cdots\!97\)\( T^{26} + \)\(89\!\cdots\!98\)\( T^{27} + \)\(24\!\cdots\!38\)\( T^{28} + \)\(11\!\cdots\!20\)\( T^{29} + \)\(23\!\cdots\!96\)\( T^{30} + \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 - 45 T + 1354 T^{2} - 29742 T^{3} + 539561 T^{4} - 8306060 T^{5} + 112851209 T^{6} - 1371437510 T^{7} + 15189795372 T^{8} - 154507375963 T^{9} + 1458900626256 T^{10} - 12844913969889 T^{11} + 106153970573282 T^{12} - 825425574095967 T^{13} + 6063417597356300 T^{14} - 42109583127809282 T^{15} + 277097438585419287 T^{16} - 1726492908240180562 T^{17} + 10192604981155940300 T^{18} - 56889155992268141607 T^{19} + \)\(29\!\cdots\!02\)\( T^{20} - \)\(14\!\cdots\!89\)\( T^{21} + \)\(69\!\cdots\!96\)\( T^{22} - \)\(30\!\cdots\!03\)\( T^{23} + \)\(12\!\cdots\!12\)\( T^{24} - \)\(44\!\cdots\!10\)\( T^{25} + \)\(15\!\cdots\!09\)\( T^{26} - \)\(45\!\cdots\!60\)\( T^{27} + \)\(12\!\cdots\!41\)\( T^{28} - \)\(27\!\cdots\!82\)\( T^{29} + \)\(51\!\cdots\!94\)\( T^{30} - \)\(69\!\cdots\!45\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 + 3 T + 319 T^{2} + 1413 T^{3} + 52260 T^{4} + 283684 T^{5} + 5926636 T^{6} + 35363672 T^{7} + 524737548 T^{8} + 3196631298 T^{9} + 38404648775 T^{10} + 228421070856 T^{11} + 2385008618577 T^{12} + 13544694897079 T^{13} + 127279992634462 T^{14} + 681564642539867 T^{15} + 5876893897353148 T^{16} + 29307279629214281 T^{17} + 235340706381120238 T^{18} + 1076898057182060053 T^{19} + 8153869850199666177 T^{20} + 33579825974933237208 T^{21} + \)\(24\!\cdots\!75\)\( T^{22} + \)\(86\!\cdots\!86\)\( T^{23} + \)\(61\!\cdots\!48\)\( T^{24} + \)\(17\!\cdots\!96\)\( T^{25} + \)\(12\!\cdots\!64\)\( T^{26} + \)\(26\!\cdots\!88\)\( T^{27} + \)\(20\!\cdots\!60\)\( T^{28} + \)\(24\!\cdots\!59\)\( T^{29} + \)\(23\!\cdots\!31\)\( T^{30} + \)\(95\!\cdots\!21\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 49 T + 1540 T^{2} - 35932 T^{3} + 690686 T^{4} - 11356182 T^{5} + 164930169 T^{6} - 2151451498 T^{7} + 25587688149 T^{8} - 280001642018 T^{9} + 2842901625207 T^{10} - 26927822341974 T^{11} + 239153197112559 T^{12} - 1997966659820780 T^{13} + 15747902890384486 T^{14} - 117290786155418928 T^{15} + 826596227322674813 T^{16} - 5512666949304689616 T^{17} + 34787117484859329574 T^{18} - \)\(20\!\cdots\!40\)\( T^{19} + \)\(11\!\cdots\!79\)\( T^{20} - \)\(61\!\cdots\!18\)\( T^{21} + \)\(30\!\cdots\!03\)\( T^{22} - \)\(14\!\cdots\!34\)\( T^{23} + \)\(60\!\cdots\!89\)\( T^{24} - \)\(24\!\cdots\!66\)\( T^{25} + \)\(86\!\cdots\!81\)\( T^{26} - \)\(28\!\cdots\!46\)\( T^{27} + \)\(80\!\cdots\!26\)\( T^{28} - \)\(19\!\cdots\!64\)\( T^{29} + \)\(39\!\cdots\!60\)\( T^{30} - \)\(59\!\cdots\!07\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 16 T + 448 T^{2} + 5762 T^{3} + 91393 T^{4} + 983901 T^{5} + 11466518 T^{6} + 104957901 T^{7} + 986290946 T^{8} + 7672516319 T^{9} + 60304654022 T^{10} + 392902225816 T^{11} + 2604421123952 T^{12} + 13876036558043 T^{13} + 79983593725448 T^{14} + 378455027503734 T^{15} + 2705248289657184 T^{16} + 20058116457697902 T^{17} + 224673914774783432 T^{18} + 2065822694651767711 T^{19} + 20550135394541900912 T^{20} + \)\(16\!\cdots\!88\)\( T^{21} + \)\(13\!\cdots\!38\)\( T^{22} + \)\(90\!\cdots\!03\)\( T^{23} + \)\(61\!\cdots\!06\)\( T^{24} + \)\(34\!\cdots\!33\)\( T^{25} + \)\(20\!\cdots\!82\)\( T^{26} + \)\(91\!\cdots\!97\)\( T^{27} + \)\(44\!\cdots\!13\)\( T^{28} + \)\(15\!\cdots\!26\)\( T^{29} + \)\(61\!\cdots\!12\)\( T^{30} + \)\(11\!\cdots\!12\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 35 T + 1021 T^{2} - 21516 T^{3} + 399391 T^{4} - 6359827 T^{5} + 92325471 T^{6} - 1214213044 T^{7} + 14857011894 T^{8} - 168822031893 T^{9} + 1807192395472 T^{10} - 18205365400176 T^{11} + 174114971113366 T^{12} - 1578868728093535 T^{13} + 13654772717571364 T^{14} - 112408551214928210 T^{15} + 884444434014014104 T^{16} - 6632104521680764390 T^{17} + 47532263829865918084 T^{18} - \)\(32\!\cdots\!65\)\( T^{19} + \)\(21\!\cdots\!26\)\( T^{20} - \)\(13\!\cdots\!24\)\( T^{21} + \)\(76\!\cdots\!52\)\( T^{22} - \)\(42\!\cdots\!67\)\( T^{23} + \)\(21\!\cdots\!74\)\( T^{24} - \)\(10\!\cdots\!16\)\( T^{25} + \)\(47\!\cdots\!71\)\( T^{26} - \)\(19\!\cdots\!93\)\( T^{27} + \)\(71\!\cdots\!71\)\( T^{28} - \)\(22\!\cdots\!64\)\( T^{29} + \)\(63\!\cdots\!81\)\( T^{30} - \)\(12\!\cdots\!65\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 11 T + 419 T^{2} + 4705 T^{3} + 92988 T^{4} + 974800 T^{5} + 13961938 T^{6} + 134959200 T^{7} + 1572046887 T^{8} + 14256371308 T^{9} + 143581697680 T^{10} + 1243450323409 T^{11} + 11282707752897 T^{12} + 94414288868823 T^{13} + 792313173380726 T^{14} + 6401161235238468 T^{15} + 50597238264220389 T^{16} + 390470835349546548 T^{17} + 2948197318149681446 T^{18} + 21430249701734313363 T^{19} + \)\(15\!\cdots\!77\)\( T^{20} + \)\(10\!\cdots\!09\)\( T^{21} + \)\(73\!\cdots\!80\)\( T^{22} + \)\(44\!\cdots\!68\)\( T^{23} + \)\(30\!\cdots\!47\)\( T^{24} + \)\(15\!\cdots\!00\)\( T^{25} + \)\(99\!\cdots\!38\)\( T^{26} + \)\(42\!\cdots\!00\)\( T^{27} + \)\(24\!\cdots\!48\)\( T^{28} + \)\(76\!\cdots\!05\)\( T^{29} + \)\(41\!\cdots\!79\)\( T^{30} + \)\(66\!\cdots\!11\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 17 T + 771 T^{2} - 11012 T^{3} + 278161 T^{4} - 3391789 T^{5} + 62501406 T^{6} - 658812412 T^{7} + 9849603202 T^{8} - 90727821928 T^{9} + 1168800833925 T^{10} - 9528902740863 T^{11} + 110551691797174 T^{12} - 813252069755460 T^{13} + 8808924372578782 T^{14} - 60086690319593739 T^{15} + 620944719838574180 T^{16} - 4025808251412780513 T^{17} + 39543261508506152398 T^{18} - \)\(24\!\cdots\!80\)\( T^{19} + \)\(22\!\cdots\!54\)\( T^{20} - \)\(12\!\cdots\!41\)\( T^{21} + \)\(10\!\cdots\!25\)\( T^{22} - \)\(54\!\cdots\!44\)\( T^{23} + \)\(39\!\cdots\!82\)\( T^{24} - \)\(17\!\cdots\!64\)\( T^{25} + \)\(11\!\cdots\!94\)\( T^{26} - \)\(41\!\cdots\!87\)\( T^{27} + \)\(22\!\cdots\!21\)\( T^{28} - \)\(60\!\cdots\!44\)\( T^{29} + \)\(28\!\cdots\!59\)\( T^{30} - \)\(41\!\cdots\!31\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 - 81 T + 3660 T^{2} - 117320 T^{3} + 2956912 T^{4} - 61849848 T^{5} + 1111681964 T^{6} - 17583755239 T^{7} + 249175830053 T^{8} - 3208020444766 T^{9} + 37964594128726 T^{10} - 417199533670751 T^{11} + 4297142430881890 T^{12} - 41845239261162310 T^{13} + 388313109620644892 T^{14} - 3456595818162838275 T^{15} + 29647910999176336857 T^{16} - \)\(24\!\cdots\!25\)\( T^{17} + \)\(19\!\cdots\!72\)\( T^{18} - \)\(14\!\cdots\!10\)\( T^{19} + \)\(10\!\cdots\!90\)\( T^{20} - \)\(75\!\cdots\!01\)\( T^{21} + \)\(48\!\cdots\!46\)\( T^{22} - \)\(29\!\cdots\!06\)\( T^{23} + \)\(16\!\cdots\!33\)\( T^{24} - \)\(80\!\cdots\!09\)\( T^{25} + \)\(36\!\cdots\!64\)\( T^{26} - \)\(14\!\cdots\!08\)\( T^{27} + \)\(48\!\cdots\!92\)\( T^{28} - \)\(13\!\cdots\!20\)\( T^{29} + \)\(30\!\cdots\!60\)\( T^{30} - \)\(47\!\cdots\!31\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 + 15 T + 582 T^{2} + 7407 T^{3} + 162930 T^{4} + 1750115 T^{5} + 28761720 T^{6} + 257676340 T^{7} + 3508004421 T^{8} + 25338350866 T^{9} + 305254006734 T^{10} + 1605840841813 T^{11} + 19003953434058 T^{12} + 49834092699619 T^{13} + 876101742410588 T^{14} - 909694021596467 T^{15} + 44804948940816447 T^{16} - 66407663576542091 T^{17} + 4668746185306023452 T^{18} + 19386309239727684523 T^{19} + \)\(53\!\cdots\!78\)\( T^{20} + \)\(33\!\cdots\!09\)\( T^{21} + \)\(46\!\cdots\!26\)\( T^{22} + \)\(27\!\cdots\!02\)\( T^{23} + \)\(28\!\cdots\!01\)\( T^{24} + \)\(15\!\cdots\!20\)\( T^{25} + \)\(12\!\cdots\!80\)\( T^{26} + \)\(54\!\cdots\!55\)\( T^{27} + \)\(37\!\cdots\!30\)\( T^{28} + \)\(12\!\cdots\!31\)\( T^{29} + \)\(71\!\cdots\!38\)\( T^{30} + \)\(13\!\cdots\!55\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 34 T + 972 T^{2} + 20901 T^{3} + 402056 T^{4} + 6757725 T^{5} + 104967360 T^{6} + 1495027893 T^{7} + 20022350636 T^{8} + 251334088386 T^{9} + 2996721539937 T^{10} + 33875654453595 T^{11} + 365933241542311 T^{12} + 3771156649534113 T^{13} + 37263605828858117 T^{14} + 352390219640645688 T^{15} + 3200138285073873965 T^{16} + 27838827351611009352 T^{17} + \)\(23\!\cdots\!97\)\( T^{18} + \)\(18\!\cdots\!07\)\( T^{19} + \)\(14\!\cdots\!91\)\( T^{20} + \)\(10\!\cdots\!05\)\( T^{21} + \)\(72\!\cdots\!77\)\( T^{22} + \)\(48\!\cdots\!74\)\( T^{23} + \)\(30\!\cdots\!96\)\( T^{24} + \)\(17\!\cdots\!67\)\( T^{25} + \)\(99\!\cdots\!60\)\( T^{26} + \)\(50\!\cdots\!75\)\( T^{27} + \)\(23\!\cdots\!96\)\( T^{28} + \)\(97\!\cdots\!39\)\( T^{29} + \)\(35\!\cdots\!32\)\( T^{30} + \)\(99\!\cdots\!66\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 39 T + 1431 T^{2} - 34708 T^{3} + 773041 T^{4} - 13965708 T^{5} + 234032966 T^{6} - 3403017682 T^{7} + 46448294726 T^{8} - 568019341368 T^{9} + 6603756562163 T^{10} - 70248493932706 T^{11} + 722668603920858 T^{12} - 6946547612092480 T^{13} + 66186053085590476 T^{14} - 603454190232013975 T^{15} + 5584537825414777508 T^{16} - 50086697789257159925 T^{17} + \)\(45\!\cdots\!64\)\( T^{18} - \)\(39\!\cdots\!60\)\( T^{19} + \)\(34\!\cdots\!18\)\( T^{20} - \)\(27\!\cdots\!58\)\( T^{21} + \)\(21\!\cdots\!47\)\( T^{22} - \)\(15\!\cdots\!36\)\( T^{23} + \)\(10\!\cdots\!66\)\( T^{24} - \)\(63\!\cdots\!46\)\( T^{25} + \)\(36\!\cdots\!34\)\( T^{26} - \)\(17\!\cdots\!36\)\( T^{27} + \)\(82\!\cdots\!01\)\( T^{28} - \)\(30\!\cdots\!04\)\( T^{29} + \)\(10\!\cdots\!99\)\( T^{30} - \)\(23\!\cdots\!73\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 32 T + 1088 T^{2} - 23137 T^{3} + 486714 T^{4} - 8120366 T^{5} + 133448243 T^{6} - 1889856135 T^{7} + 26464403740 T^{8} - 331997649864 T^{9} + 4133229432635 T^{10} - 47071798951924 T^{11} + 533593796463199 T^{12} - 5601161866848319 T^{13} + 58673100952596054 T^{14} - 572968389460590927 T^{15} + 5592319039158292684 T^{16} - 50994186661992592503 T^{17} + \)\(46\!\cdots\!34\)\( T^{18} - \)\(39\!\cdots\!11\)\( T^{19} + \)\(33\!\cdots\!59\)\( T^{20} - \)\(26\!\cdots\!76\)\( T^{21} + \)\(20\!\cdots\!35\)\( T^{22} - \)\(14\!\cdots\!56\)\( T^{23} + \)\(10\!\cdots\!40\)\( T^{24} - \)\(66\!\cdots\!15\)\( T^{25} + \)\(41\!\cdots\!43\)\( T^{26} - \)\(22\!\cdots\!74\)\( T^{27} + \)\(12\!\cdots\!94\)\( T^{28} - \)\(50\!\cdots\!53\)\( T^{29} + \)\(21\!\cdots\!08\)\( T^{30} - \)\(55\!\cdots\!68\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 + 4 T + 855 T^{2} + 4309 T^{3} + 352124 T^{4} + 1945811 T^{5} + 93690966 T^{6} + 504108188 T^{7} + 18128480187 T^{8} + 84709603850 T^{9} + 2718723343687 T^{10} + 9828561478241 T^{11} + 332212996296665 T^{12} + 837257682488933 T^{13} + 35171789290194436 T^{14} + 62777578584495400 T^{15} + 3469534574436373438 T^{16} + 6089425122696053800 T^{17} + \)\(33\!\cdots\!24\)\( T^{18} + \)\(76\!\cdots\!09\)\( T^{19} + \)\(29\!\cdots\!65\)\( T^{20} + \)\(84\!\cdots\!37\)\( T^{21} + \)\(22\!\cdots\!23\)\( T^{22} + \)\(68\!\cdots\!50\)\( T^{23} + \)\(14\!\cdots\!07\)\( T^{24} + \)\(38\!\cdots\!96\)\( T^{25} + \)\(69\!\cdots\!34\)\( T^{26} + \)\(13\!\cdots\!83\)\( T^{27} + \)\(24\!\cdots\!84\)\( T^{28} + \)\(29\!\cdots\!93\)\( T^{29} + \)\(55\!\cdots\!95\)\( T^{30} + \)\(25\!\cdots\!72\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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